Each quest will tell you a monster name, the areas you can find them, and a quantity. Note these down.
(Sometimes there will be no creatures to kill and you will instantly receive your reward)

The monsters will vary from super weak to average. If you are unable to kill a creature the quest will change after 24 hours.
(You have 24 hours to kill them). You can also have another player help, but you will need to deal the final blow.

Once you have killed the required monsters (and a few more just to be safe) return to the npc and speak about the Daily Quest again.
If you meet the requirements you will receive your reward.

You will receive a Daily Quest Token, a Daily Prize Box, and the stacking Legend Mark: "Completed Daily Quest"

The box may be opened for either an exp/ap bonus box, or an experience box. The contents will be untradable.